Craig Bryson and Eddie Malone put Keane in their shade with first-half goals but it could have been even worse for Celtic but for some key decisions and the saves of Artur Boruc.
The Hoops were a shambles in defence as Du Wei also endured a debut to forget as manager Gordon Strachan made three changes from the side which beat Hearts.
Du Wei had a shaky opening and was fortunate not to gift Clyde a shock lead in the 15th minute in a moment of controversy. He was caught off-guard as Tom Brighton raced on to the ball before coolly slotting the ball past Boruc - only for referee Kenny Clark to penalise the striker for fouling the defender in the process.
The First Division side ignored that disappointment to go close again in the 23rd minute. Steve Masterton had Boruc saving to his right to keep his right-foot effort from nestling in the top corner.
But moments later the Polish goalkeeper spilled the ball to cause panic in the Celtic goalmouth. Alex Williams rolled Malone’s cross into the back of the net only for jubilation to turn to misery again. This time the linesman had his flag up for offside.
Clyde took a deserved lead after 33 minutes when Malone crossed for Bryson to fire past Boruc from six yards, but it could have been much worse moments later. Du Wei pulled down Brighton and the referee pointed to the spot - only for Boruc to fling himself right to save Stephen O’Donnell’s effort.
But Clyde were in dreamland soon after when O’Donnell’s 37th-minute corner skimmed the head of Du Wei and Malone launched a left-foot volley into the top corner to make it 2-0.
Clyde looked like the top flight side and they would have gone further ahead in the 53rd minute but for a brilliant save from Boruc.
Substitute Aiden McGeady could have pulled the shell-shocked Parkhead man back into the game in the 63rd minute, but he failed to hit the target with keeper Peter Cherrie to beat.
Strachan’s last throw of the dice saw Maciej Zurawski introduced - and the Pole set-up a grandstand finish by hitting the back of the net with a low strike from inside the box with seven minutes remaining.
But the underdogs survived the late Celtic surge and rightly celebrated a famous and well-deserved victory.
: Cherrie, McGregor, Higgins, McKeown, Malone, Bryson (Bouadji 82), Masterton, O’Donnell, McGowan, Williams (Miller 81), Tom Brighton (Arbuckle 89).
: Boruc, Telfer, Du (Virgo 45), McManus, Wallace (McGeady 55), Nakamura, Lennon, Keane, Pearson, Hartson (Zurawski 71), Maloney.
: K Clark.